| L(s) = 1 | + (22.2 − 15.3i)3-s + (−93.2 + 53.8i)5-s + (−32.4 + 56.2i)7-s + (258. − 681. i)9-s + (−902. − 520. i)11-s + (−2.01e3 − 3.49e3i)13-s + (−1.24e3 + 2.62e3i)15-s − 322. i·17-s + 835.·19-s + (141. + 1.74e3i)21-s + (−1.15e4 + 6.69e3i)23-s + (−2.01e3 + 3.49e3i)25-s + (−4.70e3 − 1.91e4i)27-s + (−1.14e4 − 6.61e3i)29-s + (−1.83e4 − 3.17e4i)31-s + ⋯ |
| L(s) = 1 | + (0.822 − 0.568i)3-s + (−0.745 + 0.430i)5-s + (−0.0947 + 0.164i)7-s + (0.354 − 0.934i)9-s + (−0.677 − 0.391i)11-s + (−0.918 − 1.59i)13-s + (−0.369 + 0.778i)15-s − 0.0656i·17-s + 0.121·19-s + (0.0152 + 0.188i)21-s + (−0.952 + 0.550i)23-s + (−0.129 + 0.223i)25-s + (−0.239 − 0.970i)27-s + (−0.469 − 0.271i)29-s + (−0.615 − 1.06i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.239441 - 0.918313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239441 - 0.918313i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-22.2 + 15.3i)T \) |
| good | 5 | \( 1 + (93.2 - 53.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (32.4 - 56.2i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (902. + 520. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (2.01e3 + 3.49e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 322. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 835.T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.15e4 - 6.69e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.14e4 + 6.61e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.83e4 + 3.17e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.09e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-6.96e4 + 4.02e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-2.98e4 + 5.17e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.34e4 + 7.76e3i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 2.89e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-7.47e4 + 4.31e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.16e5 + 2.01e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.48e5 + 4.30e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.33e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.71e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.96e5 + 3.39e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-2.86e5 - 1.65e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.03e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.11e5 + 1.93e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95482541523274137865523434201, −12.07980993379168639804381821701, −10.70352652654293643365933915312, −9.425139211022211721312935509050, −7.901389567323228464161244985016, −7.50239304354356451559079336884, −5.70342378575133093398960333956, −3.64962126141394823615533809805, −2.49395680503349634572002794620, −0.31043019973853705479451919997,
2.17561357993257047780447044482, 3.91461879679405641636857874570, 4.88278974389370832605573163853, 7.11083086257903747960511461028, 8.182137570554976355796056104962, 9.291577391689299382389503952625, 10.32321194109950218856177001026, 11.72195782786345320169045233452, 12.80544224098190145967472725720, 14.09188362107105702014040288268
